Article
| Title: | Exponential separability is preserved by some products (English) |
| Author: | Tkachuk, Vladimir V. |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 63 |
| Issue: | 3 |
| Year: | 2022 |
| Pages: | 385-395 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions. (English) |
| Keyword: | Lindelöf space |
| Keyword: | scattered space |
| Keyword: | $\sigma$-product |
| Keyword: | function space |
| Keyword: | $P$-space |
| Keyword: | exponentially separable space |
| Keyword: | product |
| Keyword: | functionally countable space |
| Keyword: | weakly exponentially separable space |
| MSC: | 54C35 |
| MSC: | 54D65 |
| MSC: | 54G10 |
| MSC: | 54G12 |
| idZBL: | Zbl 07655808 |
| idMR: | MR4542797 |
| DOI: | 10.14712/1213-7243.2022.021 |
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| Date available: | 2023-02-13T11:08:45Z |
| Last updated: | 2023-04-20 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151541 |
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